Key Concepts in Analysis II - Vocabulary Flashcards

A set of vocabulary-style flashcards covering key theorems, definitions, and concepts from the lecture notes in Analysis II.

Derivative

The instantaneous rate of change of a function at x: f′(x) = lim_{h→0} (f(x+h) − f(x)) / h.

Chain Rule

If f(x) = F(g(x)), then (F∘g)′(x) = F′(g(x)) · g′(x).

arctan′(x)

Derivative of arctan is 1/(1 + x^2).

Mean Value Theorem (MVT)

If f is continuous on [a,b] and differentiable on (a,b), ∃ c ∈ (a,b) with f′(c) = [f(b) − f(a)]/(b − a).

Cauchy Mean Value Theorem

If f and g are continuous on [a,b] and differentiable on (a,b), ∃ c ∈ (a,b) such that f′(c)(g(b) − g(a)) = g′(c)(f(b) − f(a)).

Rolle’s Theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c ∈ (a,b) with f′(c)=0.

Taylor Theorem

For f with n+1 derivatives, f(x) = P_n(x) + f^{(n+1)}(ξ) (x−x0)^{n+1}/(n+1)! for some ξ between x0 and x.

First Derivative Test

If f′ changes sign at a point, it indicates a local max or min there.

Second Derivative Test

If f′(c)=0 and f′′(c)>0 then c is a local min; if f′′(c)<0 then a local max; if f′′(c)=0 the test is inconclusive.

Jensen’s Inequality

For a concave f, f( (t1x1+ t2x2+ t3x3) ) ≥ (t1f(x1)+ t2f(x2)+ t3f(x3)) with ti≥0, Σti=1.

Upper Darboux Sum

U(P,f) = Σ Mi Δxi with Mi = sup{f(x): x ∈ [x{i−1},x_i]}.

Lower Darboux Sum

L(P,f) = Σ mi Δxi with mi = inf{f(x): x ∈ [x{i−1},x_i]}.

Riemann Integral

∫_a^b f(x) dx = lim R(f,P,T) as the mesh of P→0, where R is a Riemann sum.

Antiderivative

A function F is an antiderivative of f if F′(x) = f(x); equivalently, ∫ f = F + C.

Fundamental Theorem of Calculus (FTC1)

If f is continuous on [a,b], F(x)=∫_a^x f(t) dt, then F′(x)=f(x).

Fundamental Theorem of Calculus (FTC2)

If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a).

Average Value of a Function

Favg = (1/(b−a)) ∫_a^b f(x) dx; the height of a rectangle with width b−a and same area as the curve.

Arc Length

Length of y=f(x) from x=a to b: L = ∫_a^b sqrt(1 + (f′(x))^2) dx.

Surface Area of Surface of Revolution

For a curve y=f(x) rotated about the x-axis: S = 2π ∫_a^b f(x) sqrt(1 + (f′(x))^2) dx.

Washer Method (Volume of Revolution)

V = π ∫_a^b [R(x)^2 − r(x)^2] dx for volumes formed by rotating around an axis.

Disk Method

Special case of the washer method with inner radius 0: V = π ∫_a^b R(x)^2 dx.

Torus Volume Formula

Volume of a torus with major radius R and minor radius r: V = 2π^2 R r^2.

Gabriel’s Horn

Rotating f(x)=1/x around the x-axis from x=1 to ∞ yields finite volume but infinite surface area.

Riemann Sums and Partitions

Approximate ∫a^b f by sums over a partition P with sample points ti; refinement leads to the integral as the limit.